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A165558 Integers that are half of their arithmetic derivatives.

%I A165558 #28 Aug 18 2015 04:55:00
%S A165558 0,16,108,729,12500,84375,3294172,9765625,22235661,2573571875,
%T A165558 678223072849,1141246682444,7703415106497,891598970659375,
%U A165558 1211500426369012,8177627877990831,234966429149994773,946484708100790625
%N A165558 Integers that are half of their arithmetic derivatives.
%C A165558 All integers of the form p^p*q^q, with q and p two distinct primes, are in the sequence. [_R. J. Mathar_, Sep 26 2009]
%C A165558 6*10^8 < a(10) <= 2573571875. a(11) <= 678223072849. [_Donovan Johnson_, Nov 03 2010]
%C A165558 By a result of Ufnarovski and Ahlander, an integer is in this sequence if and only if it has the form p^(2p) or p^p*q^q, with p and q distinct primes. See comments from A072873. [_Nathaniel Johnston_, Nov 22 2010]
%F A165558 {n: A003415(n) = 2*n}.
%e A165558 For k =84375 = 3^3*5^5, so A003415(k)/2 = 84375*(3/3+5/5)/2 = 84375 = k, which adds k=84375 to the sequence.
%p A165558 with(numtheory);
%p A165558 P:=proc(n)
%p A165558 local a,i,p,pfs;
%p A165558 for i from 1 to n do
%p A165558   pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs);  if a=2*i then print(i); fi; od;
%p A165558 end:
%p A165558 P(100000000);
%t A165558 d[0] = d[1] = 0; d[n_] := n*Total[f = FactorInteger[n]; f[[All, 2]]/f[[All, 1]] ]; Join[{0}, Reap[Do[p = Prime[n]; ip = p^(2*p); If[ip == d[ip]/2, Sow[ip]]; Do[q = Prime[k]; iq = p^p*q^q; If[iq == d[iq]/2, Sow[iq]], {k, n+1, 6}], {n, 1, 5}]][[2, 1]] // Union][[1 ;; 18]] (* _Jean-François Alcover_, Apr 22 2015, after _Nathaniel Johnston_ *)
%Y A165558 Cf. A003415, A072873.
%K A165558 nonn
%O A165558 1,2
%A A165558 _Paolo P. Lava_ and _Giorgio Balzarotti_, Sep 22 2009
%E A165558 Entries checked by _R. J. Mathar_, Sep 26 2009
%E A165558 a(7)-a(9) from _Donovan Johnson_, Nov 03 2010
%E A165558 a(10)-a(18) and general form from _Nathaniel Johnston_, Nov 22 2010